Up to 30 Chicago public schools with great ideas about how to use a longer school day this fall will receive an extra $100,000 per school under a new, privately-funded “innovation grant’’ competition announced Wednesday.

Grant winners will receive the private dollars in addition to yet unspecified stipends schools are expected to get next school year, when Mayor Rahm Emanuel and Schools CEO Jean-Claude Brizard have vowed to stretch the school day to 7 1/2 hours systemwide.

Emanuel this week stuck by his call for a 7 1/2-hour day, even though leaders of the Chicago Teachers Union and the parent group Raise Your Hand have called 7 1/2 hours too long for some families and kids.

The number is based on what’s needed “to get kids career and college ready” and is “not arbitrary,” Emmanuel said Monday.

“Many parents have asked for longer,’’ the mayor said, adding that school board members will ultimately decide the issue.

Asked if innovation grants could only be based on 7 1/2-hour days, CPS communications chief Becky Carroll would only say Thursday that “we will announce the elementary school guidelines shortly.’’ Schools “will only be able to work within the confines of what our final guidelines will state about the length of the day,” she wrote in an email.

In a news release, Brizard said he hoped the grants “will inspire schools to think creatively about how we can best deliver instruction needed to prepare our children for college and career. We want to reward those schools that create the kind of ideas that can later serve as best practices and benefit all students in CPS.’’

Grants will be awarded to schools that present “creative, clear and credible’’ visions of a “redesigned school day” that, among other things, offers a “more rigorous curriculum;’’ expands knowledge in core content areas (reading, math, science and social studies); addresses student needs, and provides both extra help and acceleration, according to the release.

Applications are due Feb. 24. Winners will be notified by late March 2012.